The sides and diagonal of the rectangle above are strictly increasing with time. At the instant when x=4 and y=3, dx/dt=dz/dt and dy/dt=k(dz/dt). What is the value of k at that instant.
solved for z with pythagorean theorem and stuck at that step. z=5
To find the value of k at the instant when x=4 and y=3, you have already found that z=5 using the Pythagorean theorem. Now let's differentiate the given equations with respect to time (t).
Given:
x = 4
y = 3
z = 5
Differentiating both sides of the equation x = 4 with respect to t:
dx/dt = 0 ... (1)
Differentiating both sides of the equation y = 3 with respect to t:
dy/dt = 0 ... (2)
Differentiating both sides of the equation z = 5 with respect to t:
dz/dt = 0 ... (3)
Now, from the given condition, we know that dx/dt = dz/dt (since these are equal) and dy/dt = k(dz/dt). Substituting the values we have:
Substituting the values into equation (1):
0 = 0 ... (4)
Substituting the values into equation (2):
0 = k(0)
0 = 0 ... (5)
Substituting the values into equation (3):
dz/dt = 0 ... (6)
From equations (4), (5), and (6), we can see that k can be any value since 0 multiplied by any number is 0. Therefore, the value of k at the instant when x=4 and y=3 is not uniquely determined.
In summary, from the given information, we cannot determine the value of k at the instant when x=4 and y=3.